fair points and good info.
There are some plugins (some free, some not) that ease the integration between the atlassian products and github. Although I think we're pretty well setup as-is. I don't know if these offer anything better for us, probably not.
(free) - Import Github issues into Jira.
https://marketplace.atlassian.com/plugins/com.atlassian.jira.plugins.jira-import ers-github-plugin
(free) - integrates git commits into jira issues.
(paid / maybe we can get for free?) - moves code review into jira as well as commit linking for issues.
https://marketplace.atlassian.com/plugins/com.xiplink.jira.git.jira_git_plugin
Regarding Github and getting data out... Github has a very extensive API that allows you to access almost anything... they are very "open data" like that. I don't think there is any data that goes into Github that one cannot get back out. The Github Issue importer plugin for example, makes use of the API to retrieve Github issues and re-format them into Jira tickets. I've not used the Github API myself.
http://developer.github.com/v3/